Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations which appear to demonstrate that such DL methods have the capacity to overcome the curse of dimensionality (COD) for PDEs in the sense that the number of computational operations they require to achieve a certain approximation accuracy $\varepsilon\in(0,\infty)$ grows at most polynomially in the PDE dimension $d\in\mathbb N$ and the reciprocal of $\varepsilon$. While there is thus far no mathematical result that proves that one of such methods is indeed capable of overcoming the COD, there are now a number of rigorous results in the literature that show that deep neural networks (DNNs) have the expressive power to approximate PDE solutions without the COD in the sense that the number of parameters used to describe the approximating DNN grows at most polynomially in both the PDE dimension $d\in\mathbb N$ and the reciprocal of the approximation accuracy $\varepsilon>0$. Roughly speaking, in the literature it is has been proved for every $T>0$ that solutions $u_d\colon [0,T]\times\mathbb R^d\to \mathbb R$, $d\in\mathbb N$, of semilinear heat PDEs with Lipschitz continuous nonlinearities can be approximated by DNNs with ReLU activation at the terminal time in the $L^2$-sense without the COD provided that the initial value functions $\mathbb R^d\ni x\mapsto u_d(0,x)\in\mathbb R$, $d\in\mathbb N$, can be approximated by ReLU DNNs without the COD. It is the key contribution of this work to generalize this result by establishing this statement in the $L^p$-sense with $p\in(0,\infty)$ and by allowing the activation function to be more general covering the ReLU, the leaky ReLU, and the softplus activation functions as special cases.

翻译：近期，已有多种用于逼近高维偏微分方程的深度学习方法被提出。这些方法在文献中引起广泛兴趣，很大程度上源于数值模拟表明：此类深度学习方法在逼近精度$\varepsilon\in(0,\infty)$下所需的计算操作次数至多以关于PDE维数$d\in\mathbb N$和$\varepsilon$倒数的多项式增长，从而具备克服维数灾难的能力。尽管尚无数学理论证明任一此类方法确实能克服维数灾难，但现有文献中已涌现一系列严谨结果，证明深度神经网络具备在不引发维数灾难的条件下逼近PDE解的表示能力——即描述逼近DNN的参数数量至多以关于PDE维数$d\in\mathbb N$和逼近精度$\varepsilon>0$倒数的多项式增长。概言之，现有文献已证明：对任意$T>0$，若初值函数$\mathbb R^d\ni x\mapsto u_d(0,x)\in\mathbb R$（$d\in\mathbb N$）可被ReLU-DNN在不引发维数灾难的条件下逼近，则具有Lipschitz连续非线性的半线性热PDE的解$u_d\colon [0,T]\times\mathbb R^d\to\mathbb R$（$d\in\mathbb N$）在终端时刻$L^2$意义下可通过ReLU-DNN在不引发维数灾难的条件下逼近。本文的核心贡献在于将该结论推广至$L^p$意义（$p\in(0,\infty)$），并将激活函数扩展至更一般情形，涵盖ReLU、leaky ReLU与softplus激活函数作为特例。